Inequalities: VJC 2024 Promo Q2

(a)

$$\begin{aligned} \frac{2x^2+3x+10}{x^2-x}&\geq2\\ \frac{2x^2+3x+10-2(x^2-x)}{x^2-x}&\geq0\\ \frac{5x+10}{x(x-1)}&\geq0 \end{aligned}$$

$\therefore -2\leq x<0\quad\text{or}\quad x>1$

(b) Replace $x$ in part (a) with $-e^x$

$-2\leq-e^x<0\quad\text{or}\quad-e^x>1$(Reject since $-e^x<0$ for all real $x$)

$0<e^x\leq2$

$x\leq\ln2\qquad\therefore\{x\in\mathbb R:x\leq\ln2\}$